1. Bragg Planes How to do a Fourier transform on paper with no calculations at all.

2. Bragg planes are always perpendicular to S   s0s0 s S -s0-s0 Since s 0 and s are the same length and have the same angle to the reflection plane, S = (s-s 0 )/ is normal to the plane.

3. The length of S is 1/d   s0s0 s S -s0-s0 The length of S is 2sin  times the lengths of s and s 0, which is 1/. So |S| = 2sin  / = 1/d  

4. 1 7 0 2 1 0 Max von Laue says: The vectors S that have amplitude > 0 are the ones where the Bragg planes all line up with the unit cell origins. This must be true for all unit cells in the crystal (ta+ub+vc) to scatter with the same phase. Using Miller indeces: S = ha*+kb*+lc*

5. d from S using Miller indeces Axes a,b,c are all orthogonal.

6. 2 1 0 Where the first Bragg plane cuts the axes The n=1 Bragg plane (normal to S at distance d) cuts the unit cell axes at 1/h1/k1/l

7. If indeces hkl are doubled, Bragg distance d is halved. 2 1 0 All unit cell origins have phase zero. But not all phase-zero Bragg planes must go through a unit cell origin. For example, the n=odd Bragg planes for the 0 2 0 reflection does not touch a single unit cell origin. 4 2 0 0 1 0 0 2 0

8. (2 3 3) Bragg planes (4 6 6) Bragg planes 3D Bragg planes Phase-zero planes intersect the cell axes at fractional coordinates (1/h,0,0), (0,1/k,0),(0,0,1/l)

9. Calculating the structure factors Draw a plane that intersects the unit cell axes at 1/h, 1/k, and 1/l (careful to consider the sign of h,k,l) Measure the phase of each atom as its distance from the nearest Bragg plane, divided by d and multiplied by 360°. Draw the scattering factor for that atom, and sum the scattering factors to get the structure factor.

10. Calculate structure factors: F( 1 1 0) F(-1 1 0) F(-2 1 0) For a unit cell with two atoms: carbon (amplitude 6) @ (0.5, 0.2, 0.0) oxygen (amplitude 8) @ (0.3, 0.4, 0.0) a b In class exercise:

11. Calculating the density Given the structure factors F(hkl), find the point(s) of maximum e-density. F(hkl) = |F(hkl)|e i  Draw Bragg planes with phase =  ( Measure phase in the direction (h,k,l) ) Erase Bragg planes with phase =  +180° After drawing and erasing all F’s, the darkest areas are the locations of the atoms.

12. Find the maximum density point given the following structure factors: F( 1 1 0) = 1 e i(108°) F(0 1 0) = 1 e i(180°) F(1 1 0) = 1 e i(-60°) F(-1 1 0) = 1 e i(60°) F(-2 1 0) = 1 e i(-10°) a b In class exercise:

13. Terms we have learned Reflection Structure factor Bragg planes Scattering factor Ewald sphere Laue conditions Reciprocal space Miller indeces Fourier transform